These two methods assume you have an HDMI capture card as converting VGA down into anything else would be a bit more challenging.

The cheapest way would probably be any dime a dozen VGA to HDMI converter you can find online (like $20-$40 maybe). My preferred way would probably be to use something like an OSSC with its VGA input to convert it to HDMI without inducing lag or degrading the quality (an OSSC is like $120).

Like Timmiluvs mentioned, there is the OSSC that can upscale/linedouble VGA to HDMI so you need that plus an HDMI capture card. Otherwie, there is the Micomsoft Xcapture-1 capturecard that has VGA on it for a more all-in-1 solution. Both are pretty expensive, the OSSC one more expensive.